In this part of the course, we shall be concerned with the behavior of physical systems under small perturbations of a known solution. The first question in this context is stability. That is, if the evolution of a system starts from an initial condition which is close to a given solution, will the system evolve towards that solution or away from it? In the former case, we call the solution stable, in the latter case unstable.
Since we are interested in stability with respect to small disturbances, it is natural to consider infinitesimal disturbances first, i.e. to linearize the system. This leads us to the study of linear stability. We shall discuss this issue in the context of systems of ordinary differential equations. We shall see that the eigenvalue problem associated with the linearized system plays a fundamental role in the study of stability. The actual stability problems encountered in fluids, of course, are partial differential equations. We shall discuss how these can be viewed as infinite systems of ordinary differential equations and treated by similar techniques. We shall point out some of the mathematical difficulties encountered in doing so, which, especially in the context of viscoelastic fluids, are not completely resolved and still a subject of ongoing research.
In the next chapter, we discuss the relationship between linear stability and nonlinear stability with respect to small disturbances.
The final chapter in this part of the course in concerned with bifurcations. Here we consider a system depending on a parameter, and a known solution which changes from stable to unstable as the parameter exceeds a critical value (for instance, in classical fluid mechanics, a steady flow may be stable for small Reynolds numbers, but unstable for large Reynolds numbers). The question then arises which new solutions may be observed when the known solution becomes unstable. In the neighborhood of the critical parameter value, it is natural to use perturbation techniques to look for new solutions which are close to the old one. If such solutions exist, we say they bifurcate from the old solution.
The exposition in these notes is, by necessity, rather limited; for instance, we largely omit proofs. There are many sources available for a more in-depth discussion of the issues we raise here, and we shall mention only a few. A discussion of stability and qualitative dynamics of systems of ordinary differential equations can be found e.g. in [8]. For introductions specifically to the topic of bifurcations, see [9,7,3,10,15]. Bifurcations with symmetry are discussed in [5,6]. A general introduction to partial differential equations is given in [12], and evolution problems associated with partial differential equations are discussed in [11]. For an introduction to classical instabilities in hydrodynamics, we refer to [4].